Ben is 3 times as old as Daniel. Twenty years ago, Ben was 8 times as old as Daniel. How old is Ben now?
Solution: We can use the given information to write down two equations that describe the ages of Ben and Daniel. Let Ben's current age be $b$ and Daniel's current age be $d$ The information in the first sentence can be expressed in the following equation: $b = 3d$ Twenty years ago, Ben was $b - 20$ years old, and Daniel was $d - 20$ years old. The information in the second sentence can be expressed in the following equation: $b - 20 = 8(d - 20)$ Now we have two independent equations, and we can solve for our two unknowns. Because we are looking for $b$ , it might be easiest to solve our first equation for $d$ and substitute it into our second equation. Solving our first equation for $d$ , we get: $d = b / 3$ . Substituting this into our second equation, we get: $b - 20 = 8($ $(b / 3)$ $- 20)$ which combines the information about $b$ from both of our original equations. Simplifying the right side of this equation, we get: $b - 20 = \dfrac{8}{3} b - 160$ Solving for $b$ , we get: $\dfrac{5}{3} b = 140$ $b = \dfrac{3}{5} \cdot 140 = 84$.